A sharp stability result for the relative isoperimetric inequality inside convex cones

TL;DR

This paper establishes a sharp stability result for the relative isoperimetric inequality within convex cones, extending classical inequalities to a more general setting with new techniques addressing the lack of translation invariance.

Contribution

It introduces a novel stability proof for the relative isoperimetric inequality inside convex cones, leveraging a variant of Gromov's method and addressing translation invariance issues.

Findings

Proves a sharp stability estimate for the inequality

Extends classical isoperimetric results to convex cones

Develops new techniques for non-translation-invariant settings

Abstract

The relative isoperimetric inequality inside an open, convex cone $\mathcal C$ states that, at fixed volume, $B_r \cap \mathcal C$ minimizes the perimeter inside $\mathcal C$. Starting from the observation that this result can be recovered as a corollary of the anisotropic isoperimetric inequality, we exploit a variant of Gromov's proof of the classical isoperimetric inequality to prove a sharp stability result for the relative isoperimetric inequality inside $\mathcal C$. Our proof follows the line of reasoning in \cite{Fi}, though several new ideas are needed in order to deal with the lack of translation invariance in our problem.